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Let $S$ be a set and let $f$ be a function from $\mathcal P(S)$ to $\mathbb R$ such that for all $A,B\in\mathcal P(S)$ with $A\cap B=\emptyset$, $$f(A\cup B)=f(A)+f(B)$$ Show that for all $A,B\in\mathcal P(S)$, $$f(A\cup B)=f(A)+f(B)−f(A\cap B)$$

I have tried to do this but I didn't quite understand.

Mahdi Klouz
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Hint: If we say $X=Y\uplus Z$ to mean that $X=Y\cup Z$ and $Y\cap Z=\emptyset$, then $$A\cup B=A\uplus(B\setminus A)\\B=(B\setminus A)\uplus(A\cap B)$$

  • Excellent use of notation! +1 – user729424 Feb 14 '20 at 22:26
  • @AndrewOstergaard LOL. I feel like there was some sort of notation similar to this for supplementary subspaces of a vector space that my professor really hated, but this was the closest thing I could find. –  Feb 14 '20 at 23:56